Let p be " Ananya is beautiful, " and let q be " Ananya is 165 centimetres tall. "
Under what conditions is the statement," Ananya is beautiful or 165 centimetres tall," false ?

Answer

To determine under what conditions the statement "Ananya is beautiful or 165 centimetres tall" is false, we will analyze the logical expression and its truth values.

### Step-by-Step Solution:

1. **Define the Statements**:
- Let \( P \) represent "Ananya is beautiful."
- Let \( Q \) represent "Ananya is 165 centimetres tall."

2. **Logical Representation**:
- The statement "Ananya is beautiful or 165 centimetres tall" can be represented as \( P \lor Q \) (where \( \lor \) denotes the logical OR).

3. **Truth Table Construction**:
- We will create a truth table to analyze the values of \( P \), \( Q \), and \( P \lor Q \).

| \( P \) | \( Q \) | \( P \lor Q \) |
|---------|---------|-----------------|
| True | True | True |
| True | False | True |
| False | True | True |
| False | False | False |

4. **Analyzing the Truth Table**:
- From the truth table, we see that \( P \lor Q \) is true in all cases except when both \( P \) and \( Q \) are false.

5. **Conclusion**:
- Therefore, the statement "Ananya is beautiful or 165 centimetres tall" is false when both \( P \) and \( Q \) are false. This means:
- Ananya is not beautiful (i.e., \( P \) is false).
- Ananya is not 165 centimetres tall (i.e., \( Q \) is false).

### Final Answer:
The statement "Ananya is beautiful or 165 centimetres tall" is false when Ananya is neither beautiful nor 165 centimetres tall.

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